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Tarski A. — Undecidable theories
Tarski A. — Undecidable theories



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Название: Undecidable theories

Автор: Tarski A.

Аннотация:

This monography consists of three papers: "A general method in
proofs of undecidability", "Undecidability and essential undecidability
in arithmetic", "Undecidability of the elementary theory of groups".


Язык: en

Рубрика: Математика/Алгебра/Математическая логика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 3d edition

Год издания: 1971

Количество страниц: 98

Добавлена в каталог: 14.05.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abelian group      19 85
Abstract projective geometry      33
Addition [+]      41. See also: Symbol of addition
Ajdukiewicz, K.      10
Antinomy of the liar      48
Arithmetic      35ff and passim
Arithmetic of integers      32ff. 39 41 67f. 77ff. 84.
Arithmetic of integers, natural numbers      30ff. 39 41 51 66.
Arithmetic of integers, positive integers      67
Arithmetic of integers, rational numbers      33. See also: Peano's arithmetic
Arithmetization of metamathematics      12
Associative system [groupoid]      86
Atomic (formula, sentence, symbol, term)      7 13 63
Axiom scheme      52f 67. Logical Non-logical
Axiom, system      10 13
Axiomatic, axiomatically built (system, theory)      10ff. 35
Axiomatizable (theory)      13ff. 19 25 31 49 60f.;
Beknays, P.      5 13 23 32 52 89
Binary operation symbol      6 20 see Symbol Symbol
Binary predicate      6 18ff. 29 34 see Symbol Symbol Symbol
Bound occurrence of a variable, binary variable      6f.
Brouwerian algebra      33
Cartesian product [$X\times Y$]      41
Center less group      85
Characteristic function [$C_p$]      46
Church, A.      4 17 19 31 35 61f. 89
Closure algebra      33
Commutative law      85
Commutative ring      70f.
Compatible (theories)      12 18f. 32 62
Complemented lattice, complemented modular lattice      33
Complete (theory)      12 14f. 19 31 61
Concatenation [operation of concatenating, $\Phi\: \hat{}\: \Psi$]      6 13 48
Conjunction sign [$\wedge$]      6
Consistent (theory)      12 15f 34 46 49f. and
Consistently interpretable      22
Constant      6 13 20ff. 41 and Logical Non-logical
Correspondence between expressions and natural numbers      12f. 46 48
Decidable (theory), decidability      3 14f. 19 31 34 62ff. 85f.
Decision problem      3f. 30 32 34f. 39f. 86;
Decision procedure      3 14 63
Deduction Theorem      6 9f. 12
Definable (function, relation, set), definability      31 40ff. 44ff. 49f. 55f. 60f
Defining, (formula)      44ff. 56
Definition      see: Possible definition Recursive
Derivable [logically derivable] (sentence), derivability      7ff. 11ff. and
Detachment [modus ponens]      see: Operation of detachment
Diagonal function [D]      40 46
Direct method (in proofs of undecidability)      3 5 39f.
Disjunction sign [$\vee$]      6
Distributive lattice      33
Divisibility relation [$|$]      78 ff.; see also: Symbol of the divisibility relation
Element relation [$\in$]      41; see also: Symbol of the element relation
Elementary theory      33
Elementary theory of Abelian groups      19 85
Elementary theory of abstract projective geometries      33
Elementary theory of Brouwerian algebras      33
Elementary theory of closure algebras      33
Elementary theory of fields      5 33 71
Elementary theory of groupoids, semigroups, etc.      86f.
Elementary theory of groups [Theory G], centerless groups, etc.      5 33 75 77 80 84ff.
Elementary theory of integral domains      70f.
Elementary theory of lattices, complemented lattices, etc.      5 33
Elementary theory of non-densely ordered rings      33 39 70ff.
Elementary theory of rings, commutative rings, etc.      5 33 39 41 70f.
Eliminating quantifiers      see: Method of eliminating quantifiers
Elimination procedure      23
equation      42 63
Equivalence sign [$\leftrightarrow$]      6
Essentially undecidable (theory)      4f. 12 14ff. 22ff. 27 30ff. 37ff. 49f. 60ff. 66ff. 71 79f. 85ff.
Existential quantifier [$\vee$]      6
Expression      6 12f. 43f. 46 48; Quantifier
Extension (of a theory)      4 11f. 14ff. and Inessential
Field      5 33 71
Finite extension (of a theory)      11 17 24
Finite group      86
Finitely axiomatizable (theory)      14 18f 22ff. 27 30ff. 39ff. 61f. 67ff. 79f. 85ff.
First-order predicate logic      see: Predicate logic
Formalized (system, theory), formalization      3ff. 33 40 44 48 66f.
Formula      7 II 13 24 42ff. and Defining Relativized
Free group      86
Free occurrence of a variable, free variable      6f. 20 43 45
Free semigroup      87
Function      41 and passim; see also: Characteristic function Diagonal Recursive
General recursive      see: Recursive
Goedel, K.      3 9 12 48 56 61 89
Group      78 85ff. Centerless Finite Free
Group operation [$\circ$]      78 see group see:
Groupoid [associative system]      86
Grzeqobczyk, A.      33 89
Hereditarily undecidable (theory)      18 32
Hilbert, D.      5 13 23 35 62 89
hold      see: Satisfaction
Identical theories      11
Identity [ = ]      5ff. 22 42 44;
Implication, implication sign [$\to$]      6 42
Incompleteness theorem      3
Inconsistent [not consistent] (theory)      28 47 50
Indirect method [interpretation method] (in proofs of undecidability)      4 30 39f.
Individual constant      6 11 20; 27;
Induction, induction principle, induction scheme      31 52f.
Inessential extension (of a theory)      11 16 22f 80 84 87
INTEGER      see: Arithmetic of integers Positive Set Symbols
Integral domain      70f.
Interpretable (theory), interpretability      4 20ff. 29ff. 34 66ff. 78; Relatively Weakly
Interpretation      4 22 30
Interpretation method      see: Indirect method
Irreducibility (of an axiom system)      40f.
Jantczax, A,      14» 89
Jaskowski, S.      33 77 89
Kalmar, L.      19 89
Kleene, S.C.      14 90
Lattice      5 33
Less-than relation [$<$]      41; see also: Symbol of the less-than relation
Logical axiom      6 7 9ff. 18
Logical consequence      8f.
Logical constant      6 8 11 22
Logically derivable      see: Derivable
Logically equivalent      25
Logically provable (sentence)      7
Logically true (sentence)      8f.
Logically valid (sentence)      7ff. 43f.
McKinsey, J.C.C.      86 90
Metamathematical dsignation (of expressions, symbols)      6 42
Metamathematical induction      53
Metamathematical symbolism      42ff.
Metamathematical variable      6
Metamathematics      3 12
Method (in proofs of undecidability)      see: Direct method Indirect Method 19 63
Model      11f. 25 66 61 63f. 72
Modular lattice      33
Mostowski, A.      13 31ff. 37 39f. 44 48 51 60f. 84 90
Multiplication [$\cdot$]      41; see also: Symbol of multiplication
Natural number      passim see Set
Negation sign [~]      6
Non-densely ordered ring, non-densely commutative ring      33 39 70ff.
Non-logical axiom      10 12ff. 18 35 41 62
Non-logical constant      6 8 llf. 18ff. 41 and
Numeral      52
Open [universal] (sentence)      86
Operation      8 41
Operation of inference      6 7 9f. 13
Operation of squaring      67 78 see
Operation symbol      6f. 20 22f. 25 27 see Unary
Operation, of concatenating, see: Concatenation; Operation of detachment [modus ponens]      7 9
Ordered commutative ring      70f.
Ordered couple [$\langle x, y \rangle$]      41
Ordered field      33
Ordered field of real numbers      71
Ordered integral domain      70f.
Ordered n-tuple [$\langle x_1, \ldots, x_n \rangle$]      41
Ordered ring      70f. see
Peano, G., Peano's arithmetic [Theory P]      4 31f. 40 61 52f. 55f. 60ff. 66
Positive integer      77
Possible definition      20ff.
Possible realization      see: Realization
Predicate logic [first-order predicate logic]      4f. 18f. 29 31 33f. 44 62 84
Predicate [relation symbol]      5 6f 20 23ff. 29 see unary
Presburger, M.      64f 90
PRODUCT      see: Cartesian product
Provable (sentence)      10
Quantifier      6f. 19 22 24 29 42 86 see Relativization Universal
Quantifier expression [$\vee x, \ \wedge x, \ \ldots$]      7
Quine, W.V.O.      7 87 90
Range (of a variable)      8 32f.
Rank (of a symbol)      6f. 20
Rational number      33
Real number      63 f.
Realization [possible realization]      8 10f. 18 20 61
Recursive (definition, procedure)      25 31 52 67
Recursive [general recursive] (function, relation, set)      3 13f. 19 21 31 35 40 44 48ff. 55f. 60f.
Recursively enumerable (set)      51 61
Regular substitution      43f.
Relation      8 41 see Element Less-than
Relation symbol      see: Predicate
Relatively interpretable (theory)      29ff. 33 87
Relatively weakly interpretable (theory)      29f. 84
Relativization of quantifiers      24 80
Relativized (formula, sentence) [$\Phi^{(p)}$]      25
Relativized (theory) [$T^{(p)}$]      24f. 29 33
Restricted decision problem      35
Ring      see: Elementary theory of rings Commutative Ordered
Ring with unit      70f
Robinson, J.      33 39f. 57 67 71 79 90
Robinson, R.M.      32f. 37 40 71 79 90
Rosser, B.      4 61 90
Ryll-Nardzewski, C.      31 55 90
Satisfaction, (to be) satisfied [(to) hold]      8 10 12 20 31
Semantical antinomies      48
Semantical notions      7f. 79
Semigroup      86f.
Sentence      7ff. 11ff. 43 and Provable Tautological True Valid
Sentential calculus      9f. 63
Sentential connectives      6f. 42
Set of integers [I]      41
Set of natural numbers [N]      41
Set theory      4 31 34
Square (of a number) [$n2^$]      56 78f.
Standard formalization      see: Theory with standard formalization
Subformula      24 26
Substitution      43f.
Subtheory      11 18f. 22 24 51ff. 60ff. and
Successor function, successor operation [S]      41; see also: Symbol of the successor operation
Symbol      passim
Symbol of addition [+]      20 30ff. 42 51f. 66f. 77f.
Symbol of identity [=]      6f. 22 42
Symbol of multiplication [$\cdot$]      30ff. 42 51f. 66f. 77
Symbol of the divisibility relation [|]      78f.
Symbol of the element relation [$\in$]      34
Symbol of the group operation [$\circ$]      77 86
Symbol of the less-than relation [$<$]      30 68
Symbol of the squaring operation [K]      67 78
Symbol of the successor operation [S]      51 66
Symbols of integers [0, 1]      30 51f. 66f. 78. Metamathematical Operation Rank Relation Technical
Szmielew, W.      19 34 40 77 85f. 90f.
Tarsxi, A.      1 3 7f. 10 12 19 22 31ff. 37 39f. 44 60f. 63 75 77 84f. 90f.
Tautological (sentence)      9f.
Technical symbol      6
Tenth problem of Hilbert      36
Term      7 42f? 52;
Theory      passim
Theory $J$, $J^<$, $J^*$, $J^+$      68ff. 77ff. 84 see
Theory $N$, $N^*$, $N^+$      30ff. 35 40 51 63 55f. 60ff. 66f. see
Theory $Q$, $Q$, $Q^*$, $Q^+$, $Q'$, $Q^x$      32ff. 3ff. 51 53f. 60ff. 66f. 87
Theory $R$, $R'$      40f. 61 52ff. 60f. 66f.
Theory F      86f.
Theory G      see: Elementary theory of groups
Theory P      see: Peano's arithmetic
Theory S      34f.
Theory with standard formalization      5 9 11 30 32 36 41 44 77. Axiomatizablo Complete Elementary Extension Formalized Inconsistent Set Undecidable
True (sentence), truth      8 30;
Unary operation symbol      6 see Symbol
Unary predicate      6 8 24f. 27 34 80
Undecidable (theory), undecidability      3ff. 14 17ff. 24 29ff 37 39 60ff. 67ff. 71 75 77 86 Essentially Hereditarily Indirect
Union (of theories)      12
Universal quantifier [$\wedge$]      6
Universal [open] (sentence)      86
Universe      8 18 63
Valid (sentence), validity      10ff. 40 46ff. 60 and
Variable      5 6ff. 11 13 20 32 44; Free Metamathematical Range
Weakly interpretable (theory)      20 21ff. 29f. 80 84;
Word problem (for groups)      36 86
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